Convergence of the MAC scheme for the compressible stationary Navier-Stokes equations

Thierry Gallouët (1) , Raphaele Herbin (1) , Jean-Claude Latché (2) , David Maltese (3)

1 I2M - Institut de Mathématiques de Marseille
2 IRSN - Institut de Radioprotection et de Sûreté Nucléaire
3 IMATH - Institut de Mathématiques de Toulon - EA 2134

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three-dimensional Cartesian grids. Existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the approximate solutions, up to a subsequence, and in an appropriate sense. We then prove that the limit of the approximate solutions satisfies the mass and momentum balance equations, as well as the equation of state, which is the main difficulty of this study.

Domaines: Analyse numérique [math.NA]

Thierry Gallouët, Raphaele Herbin, Jean-Claude Latché, David Maltese. Convergence of the MAC scheme for the compressible stationary Navier-Stokes equations. 2016. ⟨hal-01342840v1⟩